Abstract

We review the intimate connection between (super-)gravity close to a spacelike singularity(the “BKL-limit”) and the theory of Lorentzian Kac–Moody algebras. We show that in thislimit the gravitational theory can be reformulated in terms of billiard motion in a region ofhyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite)sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groupsare the Weyl groups of infinite-dimensional Kac–Moody algebras, suggesting that these algebrasyield symmetries of gravitational theories. Our presentation is aimed to be a self-contained andcomprehensive treatment of the subject, with all the relevant mathematical background materialintroduced and explained in detail. We also review attempts at making the infinite-dimensionalsymmetries manifest, through the construction of a geodesic sigma model based on a LorentzianKac–Moody algebra. An explicit example is provided for the case of the hyperbolic algebra, which is conjectured to be an underlying symmetry of M-theory. Illustrations of thisconjecture are also discussed in the context of cosmological solutions to eleven-dimensionalsupergravity.